Bond percolation of polymers

TL;DR

This study investigates bond percolation of non-interacting Gaussian polymers on a 2D lattice, revealing how the critical bond occupation fraction varies with polymer length and providing insights into percolation thresholds.

Contribution

It introduces a novel analysis of percolation thresholds for polymers of varying lengths, highlighting non-monotonic behavior of the critical density and bond fraction.

Findings

Critical bond fraction decreases with polymer length.

Critical density exhibits non-monotonic behavior.

Physical explanations involve bond occupancy and polymerization effects.

Abstract

We study bond percolation of $N$ non-interacting Gaussian polymers of $\ell$ segments on a 2D square lattice of size $L$ with reflecting boundaries. Through simulations, we find the fraction of configurations displaying {\em no} connected cluster which span from one edge to the opposite edge. From this fraction, we define a critical segment density $\rho_{c}^L(\ell)$ and the associated critical fraction of occupied bonds $p_{c}^L(\ell)$, so that they can be identified as the percolation threshold in the $L \to \infty$ limit. Whereas $p_{c}^L(\ell)$ is found to decrease monotonically with $\ell$ for a wide range of polymer lengths, $\rho_{c}^L(\ell)$ is non-monotonic. We give physical arguments for this intriguing behavior in terms of the competing effects of multiple bond occupancies and polymerization.